Integrand size = 21, antiderivative size = 54 \[ \int \frac {1}{x^{3/2} \sqrt {b \sqrt {x}+a x}} \, dx=-\frac {4 \sqrt {b \sqrt {x}+a x}}{3 b x}+\frac {8 a \sqrt {b \sqrt {x}+a x}}{3 b^2 \sqrt {x}} \]
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Time = 0.05 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2041, 2039} \[ \int \frac {1}{x^{3/2} \sqrt {b \sqrt {x}+a x}} \, dx=\frac {8 a \sqrt {a x+b \sqrt {x}}}{3 b^2 \sqrt {x}}-\frac {4 \sqrt {a x+b \sqrt {x}}}{3 b x} \]
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Rule 2039
Rule 2041
Rubi steps \begin{align*} \text {integral}& = -\frac {4 \sqrt {b \sqrt {x}+a x}}{3 b x}-\frac {(2 a) \int \frac {1}{x \sqrt {b \sqrt {x}+a x}} \, dx}{3 b} \\ & = -\frac {4 \sqrt {b \sqrt {x}+a x}}{3 b x}+\frac {8 a \sqrt {b \sqrt {x}+a x}}{3 b^2 \sqrt {x}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.65 \[ \int \frac {1}{x^{3/2} \sqrt {b \sqrt {x}+a x}} \, dx=-\frac {4 \left (b-2 a \sqrt {x}\right ) \sqrt {b \sqrt {x}+a x}}{3 b^2 x} \]
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Time = 2.13 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.76
method | result | size |
derivativedivides | \(-\frac {4 \sqrt {b \sqrt {x}+a x}}{3 b x}+\frac {8 a \sqrt {b \sqrt {x}+a x}}{3 b^{2} \sqrt {x}}\) | \(41\) |
default | \(-\frac {\sqrt {b \sqrt {x}+a x}\, \left (6 x^{\frac {5}{2}} \sqrt {b \sqrt {x}+a x}\, a^{\frac {5}{2}}+6 x^{\frac {5}{2}} a^{\frac {5}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}+3 x^{\frac {5}{2}} \ln \left (\frac {2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+2 a \sqrt {x}+b}{2 \sqrt {a}}\right ) a^{2} b -3 x^{\frac {5}{2}} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{2} b -12 x^{\frac {3}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {3}{2}}+4 \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} b \sqrt {a}\, x \right )}{3 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, b^{3} x^{\frac {5}{2}} \sqrt {a}}\) | \(194\) |
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Time = 0.44 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.54 \[ \int \frac {1}{x^{3/2} \sqrt {b \sqrt {x}+a x}} \, dx=\frac {4 \, \sqrt {a x + b \sqrt {x}} {\left (2 \, a \sqrt {x} - b\right )}}{3 \, b^{2} x} \]
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\[ \int \frac {1}{x^{3/2} \sqrt {b \sqrt {x}+a x}} \, dx=\int \frac {1}{x^{\frac {3}{2}} \sqrt {a x + b \sqrt {x}}}\, dx \]
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\[ \int \frac {1}{x^{3/2} \sqrt {b \sqrt {x}+a x}} \, dx=\int { \frac {1}{\sqrt {a x + b \sqrt {x}} x^{\frac {3}{2}}} \,d x } \]
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Time = 0.29 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.98 \[ \int \frac {1}{x^{3/2} \sqrt {b \sqrt {x}+a x}} \, dx=\frac {4 \, {\left (3 \, \sqrt {a} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} + b\right )}}{3 \, {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{3}} \]
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Timed out. \[ \int \frac {1}{x^{3/2} \sqrt {b \sqrt {x}+a x}} \, dx=\int \frac {1}{x^{3/2}\,\sqrt {a\,x+b\,\sqrt {x}}} \,d x \]
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